Percolation — Conformal Invariance & Cardy's Formula

Smirnov 2001: critical site percolation is conformally invariant — crossing probability = ₂F₁(η)

Controls

p_c ≈ 0.5928 (triangular lattice)
Cardy formula: P(η)=…

At the critical probability p_c ≈ 0.5928 for triangular lattice site percolation, the scaling limit is conformally invariant (Smirnov 2001, Fields Medal 2010).

Cardy's formula (1992) predicts the horizontal crossing probability for a rectangle of aspect ratio r:

P = Γ(2/3)/(Γ(1/3)²) · η · ₂F₁(1/3,2/3;4/3;η)

where η is the cross-ratio. This is exactly confirmed by simulations at p_c.

Colors: connected clusters. Red = left-to-right spanning cluster.